White Noise in Blender Z Pass with alpha value in Mix Shader
Switch to Branched Path Tracing

Switch the sampling method to Branched Path Tracing and set Min Transparent Bounces to 32 or higher to eliminate the noise.



Also, it is better to use Mist pass instead of Depth.



You need to map the value from 0-1 to 0-100, depending on your mist settings, to match the depth pass.



Thanks to skw Stefan Werner for the Branched Path Tracing hint. https://blenderartists.org/t/cycles-depth-and-mist-passes-for-a-transparent-material-give-a-lot-of-noise/1192751/5



Stay with Path Tracing

If you need to stay with Path Tracing, which seemingly ignores Min Transparent Bounces, you can create a new view layer with material override.





The material will be with Camera Data node and mixing by the object index N.



Change the object indexes of all the particle objects to N, easily with Copy to Selected.



It gives the same result as the mist pass.